## Normal Distribution

A specific type of a continuous distribution is the normal distribution. The normal distribution is in the shape of a bell curve and is a common occurring probability distribution in nature, or is used as a close approximation for the measurement of variables.

### Bell Curve

The normal probability density function.

$f(x)=\frac{1}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}$

The Bell curve has the following key properties:

• It is symmetrical around the y axis
• The mean of the function is zero
• The standard deviation is one.

### General Normal Probability Distribution

It is possible to translate the graph by shifting to a new mean and to change the spread of the the normal distribution by dividing our new x term by the standard deviation and multiplying the y term by the standard deviation.

$\frac{1}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}} \rightarrow \frac{1}{\sigma\sqrt{2\pi}}e{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}$

This gives us a new equation of on over sigma times the square root of two pi times e to the power of negative a half of x minus mew over sigma all squared. Where sigma is our standard deviation and mew is the the mean value.

### Spread of the Normal Distribution

The main strength of the normal distribution is that we know what percentage of the distribution lies within each standard deviation of the mean.

• $\mu-\sigma \leq X \leq \mu +\sigma$ includes about 68% of the values
• $\mu-2\sigma \leq X \leq \mu +2\sigma$ includes about 95% of the values
• $\mu-3\sigma \leq X \leq \mu +3\sigma$ includes about 99.7% of the values

### Standardised Values

To help us solve the general normal distribution, we can apply the general normal distribution to our original normal distribution. This we can do by linking our, z value, the standardise value to our x value.

$z=\frac{x-\mu}{\sigma}$

When n is positive the value lies above the mean and when n is negative the value lies below the mean. The z value can be thought of as the point on the original normal curve that incorporates the same amount of the distribution as the x point on the general normal curve.

### Practice Exam Question

Question 5 from the 2010 exam 1

Part a asks to find the probability that X is greater than five for a normally distributed random variable with a mean of five. Given that normal distributions are symmetrical we know that 50% of the distributions lie either side of the mean. Therefore the probability that we get a response greater than 5 is 0.5

Part B, asks us to find the equivalent standardise value for the probability x is greater than 7, but at the other end the distribution for when the standardised value is less than b.

Step 1 Find the value of z above which the probability is the same as x is greater than 7.

$z=\frac{x-\mu}{\sigma} \newline = \frac{7-5}{3}=\frac{2}{3}$

Given the probability of the normal distribution is symmetrical around zero, the probability that z is less than b is the same as the probability that z is greater than negative b. Therefore if we times our z value of two thirds by negative one we will get our b value.

$b=-\frac{2}{3}$

The symmetrical properties of the normal distribution are key to answering many of the normal distribution questions.